---------rc~ompmy1fri7ilgil'ifi;rt 1Cl 2001All RightsReserved

Copyright 1983 American Agricultural Economics Association

Duality does not yield a complete solution ofthe production problem and, in particuiar, itdoes not provide information needed by deci-sion makers who must make allocation deci-sions. Duality also does not yield a sufficientempirical framework for analysis of policiesrelating to inputs on specific crops, such as awheat acreage policy, unless such policies arereflected in sample data.

This paper points out some correspondinglimitations of the primal approach in its cus-tomary use. It then proposes a method to dealwith these problems. The proposed approachis closely related to earlier work of Marschakand Andrews; Mundlak and Hoch; and Zell-ner, Kmenta, and Dreze. However, it alsoconsiders the missing data problem (the allo-cations) and how to attain more efficiency inestimation.

The methodology is based on the followingassumptions that seem to characterize mostagricultural production:

(a) Allocated inputs. Most agricultural in-puts are allocated by farmers to specific pro-duction activities. For example, tractor andlabor hours, fertilizer, and pesticides are allo-cated among wheat, corn, and soybean fields.

(b) Physical constraints. Physical con-straints limit the total quantity of some inputsthat a farmer can use in a given period of time.For example, land is often available in fixedamounts in given time periods.

(c) Output determination. Output combina-tions are determined uniquely by the alloca-tion of inputs to various production activities

Richard E. Just is a professor and David Zilberman is an assistantprofessor, Department of Agricultural and Resource Economics,University of California, Berkeley. Eithan Hochman is a profes-sor, Ben-Gurion University, Beer-Sheva, Israel.

Giannini Foundation Paper No. 676. This paper was completedunder the auspices of the BARD Project No. 1-10-79.I Evidence of this practice in the general economics literature

can also be found in studies such as Powell and Gruen, andMundlak.

Perhaps the most difficult problem in estimat-ing nonexperimental agricultural productionfunctions is that input data typically are notavailable by crop. A farmer generally growsseveral crops, but the allocation of inputsamong crops is not recorded. The most popu-lar approach to this problem in recent econo-metric studies of agricultural production func-tions has been to use single-equation jointproduction functions which specify relation-ships between output quantities and aggregateinput quantities or to use corresponding rela-tionships between quantities and prices result-ing from duality under (expected) profitmaximization (Shumway and Chang, Weaver,Whittaker). I This paper investigates theapplicability of such approaches in modelingagricultural production.

Shumway, Pope, and Nash recently ad-dressed the problem of jointness in agricul-ture, concluding that various approaches areneeded depending on the source of jointness.They suggest that allocable fixed inputs are animportant source of jointness and show thatthe dual approach to production has seriouslimitations because it does not yield allocationequations (even if allocations are observed).

Key words: multioutput production, nonjointness, separability.

This paper considers whether separabilityor nonjointness is the better approach forattainingtractability formulticropproduction functionestimation. Characteristics ofagriculturalproduction associatedwith allocated inputs, physicalconstraints, and outputdetermination implysufficientnonjointnessfor estimation,whereas separabilityis lessplausible.The paper also addresses estimationof production functionswith allocatedinputswhere allocationsare not observed and demonstrates a proposed approachbywayof example.

Richard E. Just, David Zllberman, and Eithan Hochman

Estimation of Multicrop ProductionFunctions

Copyright 2001 All Rights Reserved

YIY2'" =ao(xu + X21)"" (X12 + X22)"" (X13 + X23)"'3.

The absurd implication of this relationship isthat increasing the amount of fertilizer appliedto wheat production offers the farmer a choiceof, say, either increasing wheat production orcorn production. Thus, the assumption of out-put determination cannot hold. The same istrue for (2). But farmers must allocate physi-cal inputs to distinct plots whether or not cor-responding data are recorded. Because plotsusually are used to grow only one crop at atime, these arguments imply that any relation-ship of the form

(5) f(YI' ... , YK) :::;:g(xt> ... ,xJ)lacks detail as a basis for either econometric oreconomic analysis of crop production. At aminimum, multiple-output production func-tions for agricultural crops must be consistentwith aggregation of production activities andinput allocations over different plots (crops).The absurd implication of (1) and (2) for the

agricultural case follows not from the use of ajoint production possibilities frontier [(3) also

where XkJ is the quantity of the jth inputapplied in producing the kth output. Thus, forequation (1),

where A = {au}, y = (YI' ... , Yk)', and Xkj isthe amount of input j allocated to productionof output k. If one knows how much of eachinput is allocated to each of the productionactivities (represented by columns of A), thenone can determine uniquely how much of eachoutput is produced. Hence, the choice of out-put mix is completely determined by the allo-cation of inputs. A major reason for the popu-larity of (3) is that it is consistent with thethree assumptions above.To consider the consistency of (1) and (2)

with these assumptions, suppose that farmerschoose the amounts of each input applied toeach output activity, e.g., the amount of fer-tilizer or tractor hours applied to corn land andthe amount of fertilizer or tractor hoursapplied to wheat land. In this case, the inputsin (1) and (2) must be represented as

(4)

tion of inputs among alternative productionactivities affects outputs, i.e.,

Multicrop Production Functions 771

where x and yare input and output vectors,respectively. In each case, distinct outputtrade-offs are available for a given set ofaggregate inputs.

However, for allocable inputs, a fundamen-tal difference exists between equation (3) andequations (I) and (2). The difference is thatequation (3) shows specifically how the alloca-

Ay :5 X,(3)

(1) YlY2S = aoXl""x2""X3"'3,and the constant elasticity of transformationfunction:

(2) ((>IYlc + ?>2Y2C)lIC = g(XI' X2, x3),where outputs are denoted by v and inputs byx.. A more popular model of multiple-outputtechnology in agriculture is the constraintstructure of a programming model,

A multiple-output production function is atechnical relationship that specifies possibleoutput mixes that can be produced from eachmix of inputs. Some examples include thegeneralization of the Cobb-Douglas functionproposed by Klein:

A Critique of Multiple-Output ProductionFunctions

aside from random, uncontrollable forces. Forexample, a farmer cannot change the outputmix merely by adjusting some dials once allinput allocations are determined. Alterna-tively, the mix of, say, wheat and corn pro-duced on a farm is determined by the land,fertilizer, water, labor, tractor hours, etc., thatare allocated to each enterprise.

Considering these structural features, someimplicit assumptions of traditional approachesbecome clear. Then, a more comprehensivespecification approach depending on dataavailability is proposed. Finally, a specific es-timation technique is proposed to make com-plete use of available data and a priori spec-ification when producers maximize profits andvariable input allocations are unobserved. Themethod is demonstrated with data on Israelidesert agriculture. Results show how standardaccounting data can be used to investigatevariation in management efficiency and tech-nology among farms and over time. The pa-per also demonstrates how unobserved inputallocations can be estimated.

Just, Zilberman, and Hochman

CopYright 2001 All Rights Reserved

, Note that existence of relevant Jacobians is assumed through-out the rest of this paper. Uniqueness follows under the usualassumptions of strong concavity and mono tonicity on the f.in (7). A similar point has been made by Mittelhammer, Matulich,

and Bushaw, but their analysis stops short of providing a practicalalternative approach. Their argument is that the production func-tion must be represented by multiple equations as in the generalvector function, h(y, X) = o., Here we assume that inputs are homogenous so that linear

aggregation is the only consistent method of aggregation (Green).

Y2 = .t;(X21) == A2X21".Then the choices for output mix with aggre-gate input Xl is represented by(9) (Al-1YlF,a, + (A2 -IY2F,a. - Xl = O.However, where input allocations are ob-served, econometric analysis of (8) generallyleads to much better estimates of productionfunction parameters than (9).

Moreover, when (8) is generalized such aswith two inputs in each function, then onegenerally cannot obtain equations that involveonly outputs and aggregate inputs and reflectonly production technology as in (5) and (9),i.e., that do not inc.lude additional assump-nons such as behavioral rules. To see this,!l0te that (6) and (7) ~ontain J + K equationsm J . K + J + K vanables. According to the~mpl~cit.function theorem, they can be solvedm principle for any J + K variables in terms ofthe other J . K variables (assuming the rele-va~t .Jacobians exist).! But only in very re-stnc.tIve ~ays can this lead to a relationshipn?t involving the allocations of inputs to indi-Vidual production activities." For instance,one of the resulting equations could be of theform

(10) Yk = fk(Xl, ... , Xl> Yl' ... ,Yk-l, Yk+l' ... , YK)

only if the other J + K - 1 variables excludedfrom the right-hand side were the J . K vari-abl.es representing ~nput allocations (the Xkj).This would be possible only if K = 1 or J = 1or if at least J . K - J - K + 1of the allocationvariables in X do not appear in (6) and (7).5 In~ther word~, for general agricultural applica-tions, e9u.a~lOnssuch as (5) restrict drasticallythe flexibility offered by allocation decisions.

(8)

Under some circumstances, the productionsystem in (6) and (7) can be represented by arelationship between outputs and aggregateinpu~s in the spirit of (1) and (2). For example,consider two outputs and one input with

Amer. J. Agr. Econ.

, If sufficient temporal and spatial detail is recorded in inputdata, then many inputs may apply only to a single productionactivity. Thus, ~h~ number of physical accounting relationshipsneeded for empirical purposes may depend on the detail of re-corded data.

(7) Yk = fk(Xkl, ... , XkJ), k = 1, ... , Kso that production activities are linked only bythe physical accounting relationships in (6).2Thus, under the assumptions of allocated in-puts, physical constraints, and output deter-mination, one obtains a production systemwhere nonjointness is imposed only with re-spect to inputs and not outputs (Lau).

Since this general form is not tractable formost purposes, the common approach hasb~en to assume either separability of h(y, X)w~th respect to inputs and outputs, which im-phes that hey, X) = fey) - g(X) as in (5), or toassu!fle some kind of nonjointness of the pro-duction technology (Hall, Lau). Given thatsome simplifying assumptions must be im-posed, the argument here is that allocated in-p~ts, physical constraints, and output deter-mmation better characterize agricultural pro-duction than separability.

Although the relationship in (3) is reason-able by this standard, its fixed proportions lim-itations impose nonjointness with respect toboth inputs and outputs (Lau). The approachhere is to generalize the technology matrix Acolumn by column to allow input substitutionyet re~ain additive physical accounting rela-tionships for allocable inputs. Suppose h istwice differentiable and uniquely determinesoutputs y from inputs X. Then, using this im-plicit function theorem, one can solve h(y, X)= 0 for a vector function y = f(X) assumingthe relevant Jacobian exists. Hence, if inputsare allocated among production activities (in-put Xkj affects production of Yk but not yd,one can write

j = 1, ... ,J.K

(6) L Xkj = Xj,k=l

yields a nontrivial frontier] but from the re-striction that this frontier is separable withrespect to inputs and outputs. To see thisconsider a general multi-output producticnfunction given by hey, X) = 0 where h is avector function, y is a K x 1vector of outputsand!, is K x J matrix with elements Xkj repre:senting the allocation of inputs x = (Xl' ... ,XJ)' ,

772 November /983

Co ri ht 2001 All Ri hts Reserved

To illustrate the implications of these resultsfor specification and estimation, consider the

Implications for Specification and Estimation

(13) aYk _ /3 Yk- kj--,aXkj Xkj

k = 1, 2; j = 1, 2, 3which, together with (6), is consistent with (1)for /31} = OIj and /32.1= OIj/B,j = 1,2,3. That is,all production elasticities for one product areproportional to those for the other product. Bysubstituting (13) into (11) or (12), one canverify that /311 = B/32.1'j = 1, 2, 3, for some B,and these are exactly the conditions underwhich (11) and/or (12) hold. But these restric-tions would exclude the possibility that therelative productivity of land in cotton versuscom may be higher than for fertilizer becauseof a cotton acreage restriction. Similar argu-ments can be establshed for (2). These resultsreveal that some common multi-output pro-duction functions have serious limitations foragricultural applications.

Thenk = 1,2.

this approach is also just sufficient to obtain(5) or (10) by the implicit function theorem.While first-order conditions can lead to gen-

eral forms such as (5) and (10), one also mustrecall that arbitrary specifications of (5) can beinconsistent with first-order conditions forproblems with allocation decisions. One mustinvestigate the source and implication of spec-ifications like (5) for problems with allocationdecisions. For example, two sets of implicitrestrictions which can be associated with (1)and (2) are given in (11) and (12). That is, noinput or output choices in (1) or (2) can causefailure of (11) and (12). Thus, (1) and (2) maybe applicable for some allocation problemsunder profit maximization but inappropriateunder other behavioral rules. The assumptionof profit maximization may not be unreason-able, but at least it should be recognized asimplicit in (1) and (2) in this context.

A more serious problem is that some restric-tive assumptions about technology must beimposed to obtain functions such as (1), (2), or(5). To see this for a production problem withallocable inputs, suppose that the underlyingtechnology in (7) is given by

Multicrop Production Functions 773

aYk/aXlMRTS,xI,xj:llk = / .

aYk aXj

Combining these conditions with (6) and (7)gives (K - 1) . (J - 1) + K + J equations inK + J + K J variables. Without redundancy,

where

or

(12) MRTS,xl,xj*:1I1 = MRTS,xl,xj*:Uk'k = 2, ... ,K; j* = 2, ... , J

(11) MRTv1Vk*:,xl = MRTu1Ilk*:,xj'k* = 2, ... ,K; j = 2, ... , J

In fact, use of (5) or (10) is equivalent toimposing J . K - J - K + 1 arbitrary restric-tions in addition to the technological and phys-ical relationships in (6) and (7).

If equations such as (5) and (10) are used,then the form and source of implicit additionalrestrictions should be made clear. One sourceof additional restrictions can be the behavioralrule followed by the decision maker-for ex-ample, the J . K first-order conditions of profitmaximization. In this case, the relationship in(5) or (10) no longer can be regarded as purelytechnical. Rather, it will be a reduced formcombining technical and behavioral relation-ships. Note also that first-order conditionscontain additional variables (such as prices)which also must be eliminated in reaching theright-hand side of (10).

If the first-order conditions include J . Kallocating equations with the K output pricesin P = (PI' ... , PK)' and J input prices in W =(WI' ... , wJ)', then (6) and (7) plus thefirst-order conditions constitute K + J + K . Jequations in 2(K + 1) + K J variables. Sinceprofits are homogenous in prices, one of theprice variables must be chosen as a numeraireto solve the equation system for any K + J +K . J variables. The number of restrictions isjust sufficient to find a relationship such as (5)or (10).

An alternative approach to obtaining theadditional restrictions without introducingprice variables is to use the J . K first-orderconditions for profit maxmization to solve for(K - 1) . (J - 1) profit maximization condi-tions not involving prices,

Just. Zilberman, and Hochman

-----------rC":io"'p'\\:yii'iningli"htnCJ'J2if1ioarftJl Rfg-fltsReserved

Of course, If some equations do not have stochastic distur-

With addition of stochastic disturbances, all ofthese equations can be used in estimation ofthe parameters of a multioutput productionproblem with allocations." Collectively, they

x - Xe = O.[K][1]

fiX) - y = 0

j = 2, ... ,If [(If - 1) . K]

P' af _ w.e = 0aX.j J ,

j = If+17 ... , I [I - K]

[I](18) aL-- = x - Xe = 0ac!>

where X and c!> = (c!>'r, c!>'v)' = (, X, xV, y, p, w, and Xf)' Notethat zero degree homogeneity in prices impliesthat (p, w) contains only Iv + K - 1exogenousvariables. In general, no other nonredundantequations will add more information for pa-rameter estimation.

The number of observable equations thatcan be determined from (14)-(18) depends onhow many variables are observed. Accordingto the implicit function theorem, the numberof nonredundant equations that can be ex-pressed solely with observable variables is, atmost, the number of observable variables lessI + K - 1 (the number of exogenous vari-ables). Thus, if one can find at least this manynonredundant equations that include no unob-servable data, then efficient (full information)estimation is possible.To consider a specific case of data availabil-

ity, suppose first that alldata other than shadowprices are observed. Then K + I equationsdetermining X and c!> can be discarded. Themaximum number of nonredundant equationsfor this case is attained by the system

, af , af - 0p---p--- ,aX.I aX.j

[K]aL- = fiX) - y = 0ax(17)Amer. J. Agr. Econ.

(14) aL [K]-=p-X=OayaL [Iv](15) -- = c!>v- w = 0axv

aL , af(16) -- = X -- - c!>je = 0,aX.j sx,

j = 1, ... ,I [K'I]

has first-order conditions,

Xe = x,

where x' = (x'[7 x'v) so that Xf is the If xsubvector of aggregate input uses correspond-ing to fixed or constrained inputs; Xv is asimilar Iv x 1 subvector corresponding tounconstrained variable inputs, and e = (1, 1,... , 1)'. The associated Lagrangian,

L = p'y - w'x, - X'[y - fiX)] (19)- c!>'(Xe - x),

y = fiX),

subject to

five general data groups represented by y, x,X, p,and w. The data missing in most casesare allocations in X. However, the quality andavailability of input price data (w) or aggregateinput quantity data (x) also may be limited.Thus, this section considers several cases ofdata availability. For each case, equations (6)and (7) are assumed to underlie the allocativeproblem.

First, one can estimate the production rela-tions in (7) directly if and only if data areavailable on both input allocations and output(X and y). This is the only general input alloca-tion case where technological relationshipscan be estimated without implicit assumptionsthat restrict either behavior or technology.Avoiding assumptions about behavior in pro-duction function estimation is desirable if onewishes to investigate behavioral criteria spe-cifically (Just and Pope). However, increasedefficiency in estimating production parameterscan be attained with additional nonredundantrelationships reflecting plausible behavioralrules.To consider the case where the allocations

in X are not available (or where the efficiencyassociated with behavioral specification is de-sirable), suppose the production problem isone of profit maximization,

max p'y - w'x,y,X,x,

774 November 1983

Copyright 2001 All Rights Reserved

7 Note that E is a disturbance matnx which may be diagonal.The kth diagonal element is simply a multiplicative disturbance forthe kth production function.

S The m, variable can also be called management where man-agement is regarded as a technical input following Mundlak.

where t represents time; i denotes farmer; mjis a human capital measure for farmer i: andikll ~ N(O, (Tk).8 The (Xjk are production elas-

JY,kt = n X,jktJke(3kl+Ykm,+, X) are observed.Then, the maximum number of observableequations is K + Jv (the number of observablevariables in y, x, w, and p after correcting forhomogeneity less the K + J - 1 exogenousvariables). One set of equations which reflectsfull information follows from restricted dual-ity,

JUS1, Zilberman, and Hochman

CopYright 2001 All Rights Reserved

This transformation introduces several com-plications. First, the disturbances defined by(28) and (29) have nonzero expectations. Inequation (29), the expectation of Eikl can beincluded in i3kt (or a constant term) for estima-tion. In the case of equation (28), however, aconstant term must be added in equation (24),after replacing f by r to account for the non-

I - I r U'KTn tu = n ro - Eikt + -2-'

since

(U'T-U'T)+ k K2 '

J

i3kt = f3kt + L ajk In ajk'j=J,+l

Since variable input allocations are unob-served, equation (23) provides an aggregateinput equation by summing the usual first-order conditions solved for allocations. Equa-tion (24) is obtained by taking ratios of usualfirst-order conditions, solved for allocations,since shadow prices of fixed or constrainedinputs are unobserved. Equations (25) are theusual log-linear production functions with un-observables replaced through first-order con-ditions.Equations (23)-(26) give a system which is

almost linear for estimation, given rikt. Ofcourse, actual data likelywill not fit equations(23) and (24) exactly, so random errors inprofit maximization are considered as inMarschak and Andrews; Mundlak and Hoch;or Zellner, Kmenta, and Dreze by insertingadditive random disturbances (EUtX and EUk?)in (23) and (24), E(EU{) = E(EiJk{) = O. Datatypically fit equation (26) exactly, so it can beregarded as an identity.The most difficult problem in joint estima-

tion of equations (23)-(26) is the unobserv-ability of rlkt. One possibility is to replace riktusing the equation rikt = ro + 'ikt 8lkt, where81kt = elrkT/2-fikTt - 1, E(8lkt) = O. Thus, equa-tions (23)-(26) can be rewritten replacing f byrand E by E where

Amer. J. Agr. Econ.

wherej = 1, ... ,J, [J,]

1\

XUt = L Xijkt,k=l

(26)

k = 1, ... ,K [K]

J (-)+ L ajk In 'iktj=J,+l Wjt

J,(25) In Yikt = L ajk In XUkt

J=l

k = 1, ... ,K; k '* K; j = 1, ... , J,[J, . (K - 1)]

(29)

j = J'+I' ... ,J [Jv] (27)(24) In XUkt = In ajk - In ajK

Ii -

(23) '" 'ikt + xXiJt = L ajk -- Eijt,k=l Wjt

U'kT = var(Eik{), Eik{ = ElktP+ Elkl.Following the previous section, the first-

order conditions corresponding to (14)-(18)for the case of expected profit maximizationgive 2K + J + Jv + K . J nonredundant equa-tions in 3K + 2J + Jv + K . J - 1 variables ofwhich Jv K + K + J variables (Xv, ~, and ~)are unobserved. After eliminating equationswhich determine unobserved variables (usingthe implicit function theorem), the number ofnonredundant observable equations is K + K .J, + L; One set of relationships attaining thisnumber of nonredundant equations is

where

ticities for input J m crop k, f3kt is atechnologyIweather effect for crop k at time t,and v is the effect of human capital in produc-tion of crop k. Suppose output price is randomwith Pkt = Pk(Zit) efikf', where Pk(Zit) is ex-pected output price at decision-making timebased on information set Zit and EiktP is jointlynormally distributed with Elktll and has zeromean. Thus,(22) rikt == E(Pkt Yikt) = Pk(Zit)

776 November 1983

Copyright 2001 All Rights Reserved

as its ideological basis. Each farm is privatelycontrolled, and its economic life depends onits profitability.

The data were collected from the Moshav'saccounting system supported by data on landallocation from the extension service. The al-locations of variable inputs to various cropswere not recorded. Rather, the data includedtotal expenditures or quantities of purchasedinputs. Total revenues and quantities of out-puts and land allocations were also observed.Specifically, the data were a combined cross-section and time-series yielding annual obser-vations for 1977-80 including the following:

(a) Cultivated area, production, and reve-nue by crop for tomatoes, bell peppers, on-ions, melons, and eggplants. Almost all farm-ers grow tomatoes, bell peppers, and melons.A smaller number also grow eggplants andonions.

(b) Aggregate water input and expenditure.The arid conditions and remoteness of outsidewater make it one of the most critical inputs.Drip irrigation, a very efficient way to usewater, is the dominant technology.

(c) Other purchased input expenses. Theseinclude variable costs of inputs such as fer-tilizer, pesticides, and cultivating materialsmeasured in 1980prices. For estimation, thesedata were used to construct an index of othervariable input use.

(d) Management ability. To reflect differ-ences in human capital, a Delphi panel wasassembled from farm advisors and settlementleaders and used to assess management abilityof individual farmers. The variable is a ratingfrom one to ten representing consensus of thepanel.

Other major inputs, such as labor and capi-tal, were determined by the institutional set-ting. That is, when settlements are establishedby the Jewish Agency, all farmers receive thesame machinery items, the same amount ofland, etc. Furthermore, the Moshav const-rains the amount of labor that can be hired sothat neither labor nor capital differed substan-tially over either time or farms. These condi-tions both simplified and complicated the es-timation process. Since aggregate labor andcapital use were constant within the data, noequations were needed to explain aggregatelabor and capital use. However, since alloca-tions of labor and capital also were not ob-served, it was not possible to estimate theassociated equations for the allocation oflaborand capital to the various crops. It was also

Multicrop Production Functions 777

The methodology of the previous section wasapplied to estimating production functions forindividual farmers in southern Israel. The datawere collected in a Moshav (cooperative vil-lage) in the Arava region of Israel. The Aravaincludes the plains between the Red Sea andthe Dead Sea. It is an arid region with highminimum and maximum temperatures makingit an off-season producer of winter vegetablesfor local and export markets. The Moshavconsists of seventy small family farms (10acres each) with cooperation, mutual collab-oration, and the principle of self-employment

Application to Israeli Agriculture

zero expectation. These expectations correctfor seemingly risk-averse behavior underprofit maximization discussed by Just (1975).

A second complication introduced by (27) isheteroscedasticity in the aggregate variableinput equations. This heteroscedasticity is notserious if (a) the production elasticities, theratio of revenue to the input price, and thedegree of revenue variability are similar acrosscrops; (b) revenue is highly correlated amongcrops, and variability is similar across crops;or (c) the random disturbances in revenue areswamped by errors in choosing maximizinglevels of variable inputs. A test of this hypoth-esis could not be rejected for the empiricalcase here.

A third complication is that endogenousvariables appear on the right-hand side of(23)-(26) after replacing r by r. Hence, a si-multaneous equation estimation proceduremust be used. One possibility is to ignore theinformation contributed by the equations withnonlinearities in (24) and use linear simulta-neous equations methods to estimate the re-maining equations. However, this producesinefficient estimators. Another approach is touse nonlinear two-stage or three-stage leastsquares (N2SLS or N3SLS) on the entire sys-tem. The latter approach attains consistencyand asymptotic efficiency regardless of corre-lations among disturbances in first-order con-ditions assuming homoscedasticity and nocorrelation across time and decision makers(Hausman). Note that relatively little non-linearity is included in (23)-(26) so that theusual convergence problems tend not to oc-cur. Furthermore, good starting values can begenerated by linear 2SLS estimation of equa-tions (23) and (25).

Just. Zilberman. and Hochman

-------------t:e:rOfrplVyT'iir igrrlhlTlt'll~200'1All RlgnLs Reserved

If this expression is positive, then the con-strained input is overallocated to crop K versuscrop k (and vice versa). Or, (31) can be rewrit-ten in the form of (24) as

(31)

is a measure of whether a farmer overuses orunderuses the input for any particular obser-vation. Alternatively, shift variables can beused in equation (23) to see if such misalloca-tion persists over time. For constrained in-puts, the difference in estimated marginalvalue of products is

K

XiiI - L Xiiklk=l

(30)

where ajk denotes the estimate of CXjk' Somefurther uses of the model based on these calcu-lations are to determine input misallocationsby individual farmers and to develop norms orrecommendations for input use. For example,for each unconstrained input, the magnitude of

residual crop that is relatively more land in-tensive.Both estimates show that human capital has

a significant effect on production of at leastone of the crops. The greatest significance inthe N3SLS results is for the three crops (to-matoes, bell peppers, and melons) grown bymost farmers. One explanation for why humancapital does not explain significant differencesin onion and eggplant production is that onlythe best farmers successfully produce thosecrops.Significant (neutral) technological change

was suggested for the two most importantcrops also. That is, since the constant termcorresponds to 1980, the shift terms suggest asignificant positive trend with annual weathervariations for both tomatoes and bell peppers.In addition to showing that crop-specific

production elasticities can be estimated withaggregate input data, this example also dem-onstrates other useful applications of themodel and approach. One is to estimate unob-served input allocations to crops. That is,using estimated coefficients and first-orderconditions, the unobserved allocation is esti-mated by

Amer. J. Agr. Econ.

The arguments presented here were based on numerous con-versations with farm advisors in the Arava, farmers on theMoshav itself, and Jewish Agency officials responsible for Aravasettlement.

not possible to estimate the contribution oflabor and capital to individual crops in theproduction functions.One possible approach in this case is to re-

duce further the production system in equa-tions (26)-(29) to obtain an estimable systemof equations. However, some alternative as-sumptions seemed appropriate and preservesimplicity of interpretation of the estimatedmodel. 9 That is, the tractor and equipmentseemed to have excess capacity because of thesmall acreage held by each farm. The onlyother major capital input, drip irrigationequipment, is allocated to each crop inamounts proportional to the land used. Thus,the effect of capital was assumed to bereflected in the land coefficient. Labor input isless clear, but labor prior to harvest also ap-peared to be proportional to land. Harvestlabor, on the other hand, had little effect onproduction. Family labor, supplemented byvolunteer labor, is simply adjusted so thatnone of a crop is left unharvested. Thus, anylabor effects in the production function(nonharvest labor) were assumed to bereflected in the land coefficient.Using these data, equations (23)-(26) were

estimated by nonlinear two-stage and three-stage least squares assuming constant returnsto scale. No convergence problems were ex-perienced presumably due to the model'snear-linearity. The two-stage estimator con-verged in thirty iterations, and the three-stageestimator required six iterations. The modelwas estimated using the statistical analysissystem (SAS) for less than $8 for both esti-mates combined. The results are reported intable 1. The results of the two estimationmethods were quite similar. All coefficientshave theoretically appropriate signs, and mosthave quite reasonable magnitudes. Except fortomatoes and melons, all of the water produc-tion elasticities were between .05 and .08; allof the other purchased input elasticities werebetween .22 and .44; and all implied non-purchased (land-labor-capital) productionelasticities were between .51 and .70. The to-mato and melon production functions implymuch less importance of purchased inputs.However, tomatoes are considered to be rela-tively more labor intensive, and melons to be a

778 November 1983

Copyright 2001 All Rights Reserved

This paper introduces an approach for com-prehensive consideration of multi-output pro-duction problems with the common technolog-

Conclusionswhere the latter term is the estimated constantterm of equation (24) reported in table 1 forthe random production/price case. By compar-ing the deviations in (30) and (31) with thestandard error of the disturbance estimated forthe respective equations (23) and (24), one can

see whether each farmer is within a confidenceinterval around estimated profit-maximizationrelationships.

Two-Stage Three-StageLeast Squares Least Squares

Equation/coefficient Estimate r-Ratio Estimate t-Ratio

Tomato production functionConstant term 7.545 10.56 8.623 13.27Water elasticity 0.0374 2.78 0.0214 1.91Other purchased input elasticity 0.0865 1.49 0.0173 0.34Management 0.0585 1.73 0.0377 1.311977 shift -0.2234 -2.52 -0.2248 -2.951978 shift -0.0398 -0.46 -0.0094 -0.131979 shift -0.2066 -2.47 -0.2732 -3.82

Bell pepper production functionConstant term 4.679 12.94 4.534 13.64Water elasticity 0.0457 7.14 0.0509 9.32Other purchased input elasticity 0.2905 9.12 0.3087 11.01Management 0.0583 2.86 0.0246 1.451977 shift -0.5084 -10.39 -0.5505 -13.231978 shift -0.5470 -11.44 -0.5175 -12.611979 shift -0.2839 -5.64 -0.2909 -6.84

Onion production functionConstant term 2.888 2.79 2.496 2.49Water elasticity 0.0509 3.49 0.0551 4.28Other purchased input elasticity 0.3990 4.34 0.4309 5.04Management 0.0409 1.01 0.0200 0.611977 shift 0.3757 4.04 0.4975 7.231978 shift 0.3426 2.23 0.3768 3.131979 shift 0.3722 3.32 0.4463 5.14

Melon production functionConstant term 6.368 22.40 6.196 24.87Water elasticity 0.0049 1.29 0.0085 2.64Other purchased input elasticity 0.0714 3.52 0.0807 4.39Management 0.0507 1.03 0.0624 1.731977 shift -0.0844 -0.64 0.0329 0.341978 shift 0.2826 2.66 0.2060 2.671979 shift -0.0391 -0.33 -0.0083 -0.10

Eggplant production functionConstant term 4.817 7.52 4.840 7.83Water elasticity 0.0788 8.64 0.0776 9.57Other purchased input elasticity 0.2225 4.18 0.2233 4.51Management 0.0849 2.21 0.0310 0.961977 shift -0.0858 -0.75 -0.0172 -0.181978 shift 0.1203 1.18 0.3128 3.661979 shift -0.2396 -2.04 0.0305 0.33

Constant term of equation (24)k, bell peppers; K, tomatoes 2.0264 16.39 2.1984 21.29k, onions; K, tomatoes 2.0776 9.11 2.3160 10.74k, melons; K, tomatoes 1.9227 15.57 2.0937 19.97k, eggplants; K, tomatoes 1.0054 7.68 1.1093 8.92

Table 1. Estimated Model

Multicrop Production Functions 779Just, Zilberman, and Hochman

--------rc7\o"'py"'r:;;;lgihh'flf ir'\c 2mnAIf RightsReserved

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References

ical characteristics and data availability facedin agriculture. It uses aU available informationfrom both-technological and behavior assump-tions in producing estimates of multi-outputproduction functions where allocations ofvariable inputs among crops are unobserved.The empirical results show that the approachis practical and inexpensive and provides rea-sonable estimates.The results also show that the popular

single-equation, multiple-output productionfunction approach is relatively unattractive bycomparison. First, single-equation, multiple-output production functions impose just asmany restrictions as this approach (eventhough the restrictions are not recognizedexplicitly). Second, the related information isnot exploited for estimation. Finally, the re-sults are not practical since the effect of inputallocations cannot be estimated. Hence, theresults cannot be used in analyzing input-allocation decisions or making related recom-mendations.The approach presented here has wide po-

tential for examining agricultural productionproblems. For example, by varying the tech-nological specification, one can examine non-neutrality in outputs as well as inputs. Theestimates in table 1 suggest that the produc-tion-possibilities curve for, say, bell peppersand melons has been shifting in favor of bellpeppers. In addition, by extending the humancapital specification, one can examine the rela-tive effects of farmers' worker ability versusallocative ability. One approach would be toregress squared estimated disturbances infirst-order conditions on human capital vari-ables. Similarly, the approach developed foradapting specification to data observability of-fers rich possibilities for empirical investiga-tion of other unexplored issues.

[Received March 1983; revision acceptedMay 1983.]

780 November 1983